# Tagged Questions

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### Not lebesgue integrable function?

I want to consider the function $f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$ And I have ...
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### lebesgue integral of $f(x^n)$

I know that $f:[0,1]\to \mathbb{R}$ is continuous at $0$, and $f\in L_1([0,1])$. How can one prove that $f(x^n)\in L_1([0,1])$, for any $n\in \mathbb{N}$?
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### For which a and b is $\int_0^{1/2} r^{a+n-1}|\log(r)|^b dr<\infty$?

The problem I am working on asks which real values of a and b make $|x|^a|\log|x||^b$ integrable over $\{x \in \mathbb{R}: |x| < 1/2\}$, but I reinterpret the question to asking which real values ...
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### f being a Lebesgue integrable function on $(0, a)$ implies that $g(x) = \int_x^a (f(t)/t)dt$ is also integrable.

I need to prove: If f is Lebesgue integrable on $(0, a)$ and $g(x) = \int_x^a (f(t)/t)dt$, then g is integrable on $(0, a)$. I know that since f is integrable on the interval $(0, a)$ I have ...
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### Integral of product of two measurable functions

I have to show that for $\psi$ and $\phi$ two positive and measurable functions: $$\int_X \phi\psi \, d\mu \le \sqrt{\int_X \phi^2 \, d\mu} \sqrt{\int_X \psi^2 \, d\mu}$$ I know that for $f,g$ two ...
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### Is this function Lebesgue integrable? [duplicate]

I have to decice if the following function is Lebesgue-integrable on $[0,1]$: $$g(x)=\frac{1}x\cos\left(\frac{1}x\right)$$ where $x\in[0,1]$. $g(x)$ is Lebesgue integrable if and only if the ...
Let $m$ be a probability measure on $\mathbb{R}^n$, so that $m(\mathbb{R}^n) = 1$. Consider two measurable functions $f: \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$, and $g : ... 1answer 42 views ###$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$for$f\in L^{p}$,$p \in [1,\infty)$For$f\in L^{p}$,$p \in [1,\infty)$we want to prove: $$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$ I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ... 1answer 102 views ### Evaluating a Gaussian Integral How to prove that $$\int_{\mathbb{R}^N}e^{-\langle Ax,x\rangle}\operatorname{dm}(x)=\left(\frac{\pi^N}{\det A}\right)^{\frac{1}{2}}$$ Where$A:\mathbb{R}^{N}\to\mathbb{R}^{N}$is a symmetric ... 0answers 74 views ### Show derivative of integral equals integral of partial derivative if M[0,1]-measurable I am trying to determine a method of approaching the following: Suppose that$f:[0,1] \times (0,1)\rightarrow\mathbb{R}$is such that, for each$y \in (0,1)$, the function$f^{[y]}(x) = f(x,y)$... 2answers 93 views ### prove$ e^{bt} \ \int_0^t \ f(s) ds=\int_0^t \ ( e^{-bs} \ f(s)-be^{-bs}\int_0^s\ f(u)\ du) \ ds \$ [duplicate]
please How can I prove that $$e^{-bt} \ \int_0^t \ f(s) ds=\int_0^t \ ( e^{-bs} \ f(s)-be^{-bs}\int_0^s\ f(u)\ du) \ ds$$ f non-negative measurable function I would appreciate it enormously if ...